The problem is all these names and classifications are made up by guitarists, who want to simplify everything.
Minihumbucker is a form factor. Both the (vintage) Epiphone/Les Paul Deluxe minihumbuckers and Firebird pickups are minihumbuckers...two series coils that are RW/RP from each other. WHen Gibson designed the Firebird, they want to cut costs, they the Firebird PUs were actually designed to be the cost-cutting replacement for the Epiphone minihumbucker.
The reality is that there are multiple types of minihumbuckers, especially if you throw in hybirds made by Seymour Duncan, DiMarzio, and others.
For the OP, Les Paul Deluxe minihuckers fall somewhere inbetween PAF and Strat in tone and output. They were "rounder" and more balanced than the Firebird PU. Firebird PUs had a more aggressive top end, and inherent distorted lo-fi "rasp", just listen to Johnny Winter. Once you go thrid market, anything can happen, for instance, Seymour Duncan minihumbuckers look like Firebird PUs, but are built more like Epiphone minihumbuckers under the cover.
This, 100%.
Similar to the way P90's and Fender-style single-coil pickups are both single-coil pickups, yet their magnet and pole construction are very different.
I'm going to respectfully disagree with these claims, at least historically. The Mini-Humbucker and Firebird were known by two distinct names originally.
But guitarists did mix the names
due to the form factor. "Oh look, it's smaller and it's supposed to be a humbucker. It must be a mini-humbucker!". Then pickup manufacturers began to use the terms incorrectly (I'm lookin' at you, Seymour). Now it's become non-distinct. Epiphone doing that is just embarrassing - especially when the FB720 is supposed to closely model an early-ish Gibson Firebird in construction.
A mini-humbucker is still a humbucker with exposed steel pole pieces in one side, a steel slug in the other side, and a magnet below. A Firebird is still a bar magnet in each coil with a closed cover.
Unfortunatly the mixing of nomenclature forces you to actually research what's under the hood of the pickup cover in order to know which sound you're going to get.
As for
@Heartbreaker_Esq's quest, I usually suggest the Mini. As others noted, the Firebird is
very bright. One in the neck is like a Tele bridge pickup in the neck, without hum. If that's what you want, it's your huckleberry. A mini will give you mellow humbucker (think BB King) when dialed back just a hair on the tone; be as bright as the typical single coil all the way up, and sing like a humbucker with a little gain on the amp or a pedal.
I think you'll want something high output in the bridge to compete with it.